statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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rpow_le_one_of_one_le_of_neg {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (hz : z < 0) : x^z ≤ 1 | begin
cases x,
{ simp [top_rpow_of_neg hz, zero_lt_one] },
{ simp only [one_le_coe_iff, some_eq_coe] at hx,
simp [coe_rpow_of_ne_zero (ne_of_gt (lt_of_lt_of_le zero_lt_one hx)),
nnreal.rpow_le_one_of_one_le_of_nonpos hx (le_of_lt hz)] },
end | lemma | ennreal.rpow_le_one_of_one_le_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.rpow_le_one_of_one_le_of_nonpos",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_rpow {x : ℝ≥0∞} {z : ℝ} (hx : 1 < x) (hz : 0 < z) : 1 < x^z | begin
cases x,
{ simp [top_rpow_of_pos hz] },
{ simp only [some_eq_coe, one_lt_coe_iff] at hx,
simp [coe_rpow_of_nonneg _ (le_of_lt hz), nnreal.one_lt_rpow hx hz] }
end | lemma | ennreal.one_lt_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.one_lt_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_rpow {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (hz : 0 < z) : 1 ≤ x^z | begin
cases x,
{ simp [top_rpow_of_pos hz] },
{ simp only [one_le_coe_iff, some_eq_coe] at hx,
simp [coe_rpow_of_nonneg _ (le_of_lt hz), nnreal.one_le_rpow hx (le_of_lt hz)] },
end | lemma | ennreal.one_le_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.one_le_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_rpow_of_pos_of_lt_one_of_neg {x : ℝ≥0∞} {z : ℝ} (hx1 : 0 < x) (hx2 : x < 1)
(hz : z < 0) : 1 < x^z | begin
lift x to ℝ≥0 using ne_of_lt (lt_of_lt_of_le hx2 le_top),
simp only [coe_lt_one_iff, coe_pos] at ⊢ hx1 hx2,
simp [coe_rpow_of_ne_zero (ne_of_gt hx1), nnreal.one_lt_rpow_of_pos_of_lt_one_of_neg hx1 hx2 hz],
end | lemma | ennreal.one_lt_rpow_of_pos_of_lt_one_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"le_top",
"lift",
"nnreal.one_lt_rpow_of_pos_of_lt_one_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_rpow_of_pos_of_le_one_of_neg {x : ℝ≥0∞} {z : ℝ} (hx1 : 0 < x) (hx2 : x ≤ 1)
(hz : z < 0) : 1 ≤ x^z | begin
lift x to ℝ≥0 using ne_of_lt (lt_of_le_of_lt hx2 coe_lt_top),
simp only [coe_le_one_iff, coe_pos] at ⊢ hx1 hx2,
simp [coe_rpow_of_ne_zero (ne_of_gt hx1),
nnreal.one_le_rpow_of_pos_of_le_one_of_nonpos hx1 hx2 (le_of_lt hz)],
end | lemma | ennreal.one_le_rpow_of_pos_of_le_one_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"lift",
"nnreal.one_le_rpow_of_pos_of_le_one_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nnreal_rpow (x : ℝ≥0∞) (z : ℝ) : (x.to_nnreal) ^ z = (x ^ z).to_nnreal | begin
rcases lt_trichotomy z 0 with H|H|H,
{ cases x, { simp [H, ne_of_lt] },
by_cases hx : x = 0,
{ simp [hx, H, ne_of_lt] },
{ simp [coe_rpow_of_ne_zero hx] } },
{ simp [H] },
{ cases x, { simp [H, ne_of_gt] },
simp [coe_rpow_of_nonneg _ (le_of_lt H)] }
end | lemma | ennreal.to_nnreal_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_rpow (x : ℝ≥0∞) (z : ℝ) : (x.to_real) ^ z = (x ^ z).to_real | by rw [ennreal.to_real, ennreal.to_real, ←nnreal.coe_rpow, ennreal.to_nnreal_rpow] | lemma | ennreal.to_real_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.to_nnreal_rpow",
"ennreal.to_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_rpow_of_pos {x p : ℝ} (hx_pos : 0 < x) :
ennreal.of_real x ^ p = ennreal.of_real (x ^ p) | begin
simp_rw ennreal.of_real,
rw [coe_rpow_of_ne_zero, coe_eq_coe, real.to_nnreal_rpow_of_nonneg hx_pos.le],
simp [hx_pos],
end | lemma | ennreal.of_real_rpow_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.of_real",
"real.to_nnreal_rpow_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_rpow_of_nonneg {x p : ℝ} (hx_nonneg : 0 ≤ x) (hp_nonneg : 0 ≤ p) :
ennreal.of_real x ^ p = ennreal.of_real (x ^ p) | begin
by_cases hp0 : p = 0,
{ simp [hp0], },
by_cases hx0 : x = 0,
{ rw ← ne.def at hp0,
have hp_pos : 0 < p := lt_of_le_of_ne hp_nonneg hp0.symm,
simp [hx0, hp_pos, hp_pos.ne.symm], },
rw ← ne.def at hx0,
exact of_real_rpow_of_pos (hx_nonneg.lt_of_ne hx0.symm),
end | lemma | ennreal.of_real_rpow_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.of_real"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_left_injective {x : ℝ} (hx : x ≠ 0) :
function.injective (λ y : ℝ≥0∞, y^x) | begin
intros y z hyz,
dsimp only at hyz,
rw [←rpow_one y, ←rpow_one z, ←_root_.mul_inv_cancel hx, rpow_mul, rpow_mul, hyz],
end | lemma | ennreal.rpow_left_injective | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_left_surjective {x : ℝ} (hx : x ≠ 0) :
function.surjective (λ y : ℝ≥0∞, y^x) | λ y, ⟨y ^ x⁻¹, by simp_rw [←rpow_mul, _root_.inv_mul_cancel hx, rpow_one]⟩ | lemma | ennreal.rpow_left_surjective | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_left_bijective {x : ℝ} (hx : x ≠ 0) :
function.bijective (λ y : ℝ≥0∞, y^x) | ⟨rpow_left_injective hx, rpow_left_surjective hx⟩ | lemma | ennreal.rpow_left_bijective | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnrpow_pos (a : ℝ≥0) (b : ℝ) (b' : ℕ) (c : ℝ≥0)
(hb : b = b') (h : a ^ b' = c) : a ^ b = c | by rw [← h, hb, nnreal.rpow_nat_cast] | theorem | norm_num.nnrpow_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.rpow_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnrpow_neg (a : ℝ≥0) (b : ℝ) (b' : ℕ) (c c' : ℝ≥0)
(hb : b = b') (h : a ^ b' = c) (hc : c⁻¹ = c') : a ^ -b = c' | by rw [← hc, ← h, hb, nnreal.rpow_neg, nnreal.rpow_nat_cast] | theorem | norm_num.nnrpow_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.rpow_nat_cast",
"nnreal.rpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ennrpow_pos (a : ℝ≥0∞) (b : ℝ) (b' : ℕ) (c : ℝ≥0∞)
(hb : b = b') (h : a ^ b' = c) : a ^ b = c | by rw [← h, hb, ennreal.rpow_nat_cast] | theorem | norm_num.ennrpow_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.rpow_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ennrpow_neg (a : ℝ≥0∞) (b : ℝ) (b' : ℕ) (c c' : ℝ≥0∞)
(hb : b = b') (h : a ^ b' = c) (hc : c⁻¹ = c') : a ^ -b = c' | by rw [← hc, ← h, hb, ennreal.rpow_neg, ennreal.rpow_nat_cast] | theorem | norm_num.ennrpow_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.rpow_nat_cast",
"ennreal.rpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prove_nnrpow : expr → expr → tactic (expr × expr) | prove_rpow' ``nnrpow_pos ``nnrpow_neg ``nnreal.rpow_zero `(ℝ≥0) `(ℝ) `(1:ℝ≥0) | def | norm_num.prove_nnrpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.rpow_zero"
] | Evaluate `nnreal.rpow a b` where `a` is a rational numeral and `b` is an integer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prove_ennrpow : expr → expr → tactic (expr × expr) | prove_rpow' ``ennrpow_pos ``ennrpow_neg ``ennreal.rpow_zero `(ℝ≥0∞) `(ℝ) `(1:ℝ≥0∞) | def | norm_num.prove_ennrpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.rpow_zero"
] | Evaluate `ennreal.rpow a b` where `a` is a rational numeral and `b` is an integer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_nnrpow_ennrpow : expr → tactic (expr × expr) | | `(@has_pow.pow _ _ nnreal.real.has_pow %%a %%b) := b.to_int >> prove_nnrpow a b
| `(nnreal.rpow %%a %%b) := b.to_int >> prove_nnrpow a b
| `(@has_pow.pow _ _ ennreal.real.has_pow %%a %%b) := b.to_int >> prove_ennrpow a b
| `(ennreal.rpow %%a %%b) := b.to_int >> prove_ennrpow a b
| _ := tactic.failed | def | norm_num.eval_nnrpow_ennrpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | Evaluates expressions of the form `rpow a b` and `a ^ b` in the special case where
`b` is an integer and `a` is a positive rational (so it's really just a rational power). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnrpow_pos {a : ℝ≥0} (ha : 0 < a) (b : ℝ) : 0 < a ^ b | nnreal.rpow_pos ha | lemma | tactic.positivity.nnrpow_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"nnreal.rpow_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prove_nnrpow (a b : expr) : tactic strictness | do
strictness_a ← core a,
match strictness_a with
| positive p := positive <$> mk_app ``nnrpow_pos [p, b]
| _ := failed -- We already know `0 ≤ x` for all `x : ℝ≥0`
end | def | tactic.positivity.prove_nnrpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | Auxiliary definition for the `positivity` tactic to handle real powers of nonnegative reals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ennrpow_pos {a : ℝ≥0∞} {b : ℝ} (ha : 0 < a) (hb : 0 < b) : 0 < a ^ b | ennreal.rpow_pos_of_nonneg ha hb.le | lemma | tactic.positivity.ennrpow_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.rpow_pos_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prove_ennrpow (a b : expr) : tactic strictness | do
strictness_a ← core a,
strictness_b ← core b,
match strictness_a, strictness_b with
| positive pa, positive pb := positive <$> mk_app ``ennrpow_pos [pa, pb]
| positive pa, nonnegative pb := positive <$> mk_app ``ennreal.rpow_pos_of_nonneg [pa, pb]
| _, _ := failed -- We already know `0 ≤ x` for all `x : ... | def | tactic.positivity.prove_ennrpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [
"ennreal.rpow_pos_of_nonneg"
] | Auxiliary definition for the `positivity` tactic to handle real powers of extended nonnegative
reals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
positivity_nnrpow_ennrpow : expr → tactic strictness | | `(@has_pow.pow _ _ nnreal.real.has_pow %%a %%b) := prove_nnrpow a b
| `(nnreal.rpow %%a %%b) := prove_nnrpow a b
| `(@has_pow.pow _ _ ennreal.real.has_pow %%a %%b) := prove_ennrpow a b
| `(ennreal.rpow %%a %%b) := prove_ennrpow a b
| _ := failed | def | tactic.positivity_nnrpow_ennrpow | analysis.special_functions.pow | src/analysis/special_functions/pow/nnreal.lean | [
"analysis.special_functions.pow.real"
] | [] | Extension for the `positivity` tactic: exponentiation by a real number is nonnegative when the
base is nonnegative and positive when the base is positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow (x y : ℝ) | ((x : ℂ) ^ (y : ℂ)).re | def | real.rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | The real power function `x ^ y`, defined as the real part of the complex power function.
For `x > 0`, it is equal to `exp (y log x)`. For `x = 0`, one sets `0 ^ 0=1` and `0 ^ y=0` for
`y ≠ 0`. For `x < 0`, the definition is somewhat arbitary as it depends on the choice of a complex
determination of the logarithm. With ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_eq_pow (x y : ℝ) : rpow x y = x ^ y | rfl | lemma | real.rpow_eq_pow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_def (x y : ℝ) : x ^ y = ((x : ℂ) ^ (y : ℂ)).re | rfl | lemma | real.rpow_def | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_def_of_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : x ^ y =
if x = 0
then if y = 0
then 1
else 0
else exp (log x * y) | by simp only [rpow_def, complex.cpow_def];
split_ifs;
simp [*, (complex.of_real_log hx).symm, -complex.of_real_mul, -is_R_or_C.of_real_mul,
(complex.of_real_mul _ _).symm, complex.exp_of_real_re] at * | lemma | real.rpow_def_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"complex.cpow_def",
"complex.exp_of_real_re",
"complex.of_real_log",
"complex.of_real_mul",
"exp",
"is_R_or_C.of_real_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_def_of_pos {x : ℝ} (hx : 0 < x) (y : ℝ) : x ^ y = exp (log x * y) | by rw [rpow_def_of_nonneg (le_of_lt hx), if_neg (ne_of_gt hx)] | lemma | real.rpow_def_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_mul (x y : ℝ) : exp (x * y) = (exp x) ^ y | by rw [rpow_def_of_pos (exp_pos _), log_exp] | lemma | real.exp_mul | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_one_rpow (x : ℝ) : exp 1 ^ x = exp x | by rw [←exp_mul, one_mul] | lemma | real.exp_one_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"exp",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_eq_zero_iff_of_nonneg {x y : ℝ} (hx : 0 ≤ x) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 | by { simp only [rpow_def_of_nonneg hx], split_ifs; simp [*, exp_ne_zero] } | lemma | real.rpow_eq_zero_iff_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_def_of_neg {x : ℝ} (hx : x < 0) (y : ℝ) : x ^ y = exp (log x * y) * cos (y * π) | begin
rw [rpow_def, complex.cpow_def, if_neg],
have : complex.log x * y = ↑(log(-x) * y) + ↑(y * π) * complex.I,
{ simp only [complex.log, abs_of_neg hx, complex.arg_of_real_of_neg hx,
complex.abs_of_real, complex.of_real_mul], ring },
{ rw [this, complex.exp_add_mul_I, ← complex.of_real_exp, ← complex.of... | lemma | real.rpow_def_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"abs_of_neg",
"complex.I",
"complex.I_re",
"complex.abs_of_real",
"complex.add_re",
"complex.arg_of_real_of_neg",
"complex.cpow_def",
"complex.exp_add_mul_I",
"complex.log",
"complex.mul_re",
"complex.of_real_cos",
"complex.of_real_eq_zero",
"complex.of_real_exp",
"complex.of_real_im",
"... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_def_of_nonpos {x : ℝ} (hx : x ≤ 0) (y : ℝ) : x ^ y =
if x = 0
then if y = 0
then 1
else 0
else exp (log x * y) * cos (y * π) | by split_ifs; simp [rpow_def, *]; exact rpow_def_of_neg (lt_of_le_of_ne hx h) _ | lemma | real.rpow_def_of_nonpos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_pos_of_pos {x : ℝ} (hx : 0 < x) (y : ℝ) : 0 < x ^ y | by rw rpow_def_of_pos hx; apply exp_pos | lemma | real.rpow_pos_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_zero (x : ℝ) : x ^ (0 : ℝ) = 1 | by simp [rpow_def] | lemma | real.rpow_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ) ^ x = 0 | by simp [rpow_def, *] | lemma | real.zero_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_rpow_eq_iff {x : ℝ} {a : ℝ} : 0 ^ x = a ↔ (x ≠ 0 ∧ a = 0) ∨ (x = 0 ∧ a = 1) | begin
split,
{ intros hyp,
simp only [rpow_def, complex.of_real_zero] at hyp,
by_cases x = 0,
{ subst h,
simp only [complex.one_re, complex.of_real_zero, complex.cpow_zero] at hyp,
exact or.inr ⟨rfl, hyp.symm⟩},
{ rw complex.zero_cpow (complex.of_real_ne_zero.mpr h) at hyp,
exact o... | lemma | real.zero_rpow_eq_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"complex.cpow_zero",
"complex.of_real_zero",
"complex.one_re",
"complex.zero_cpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_rpow_iff {x : ℝ} {a : ℝ} : a = 0 ^ x ↔ (x ≠ 0 ∧ a = 0) ∨ (x = 0 ∧ a = 1) | by rw [←zero_rpow_eq_iff, eq_comm] | lemma | real.eq_zero_rpow_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_one (x : ℝ) : x ^ (1 : ℝ) = x | by simp [rpow_def] | lemma | real.rpow_one | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_rpow (x : ℝ) : (1 : ℝ) ^ x = 1 | by simp [rpow_def] | lemma | real.one_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_rpow_le_one (x : ℝ) : (0 : ℝ) ^ x ≤ 1 | by { by_cases h : x = 0; simp [h, zero_le_one] } | lemma | real.zero_rpow_le_one | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_rpow_nonneg (x : ℝ) : 0 ≤ (0 : ℝ) ^ x | by { by_cases h : x = 0; simp [h, zero_le_one] } | lemma | real.zero_rpow_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_nonneg_of_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : 0 ≤ x ^ y | by rw [rpow_def_of_nonneg hx];
split_ifs; simp only [zero_le_one, le_refl, le_of_lt (exp_pos _)] | lemma | real.rpow_nonneg_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_rpow_of_nonneg {x y : ℝ} (hx_nonneg : 0 ≤ x) : |x ^ y| = |x| ^ y | begin
have h_rpow_nonneg : 0 ≤ x ^ y, from real.rpow_nonneg_of_nonneg hx_nonneg _,
rw [abs_eq_self.mpr hx_nonneg, abs_eq_self.mpr h_rpow_nonneg],
end | lemma | real.abs_rpow_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.rpow_nonneg_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_rpow_le_abs_rpow (x y : ℝ) : |x ^ y| ≤ |x| ^ y | begin
cases le_or_lt 0 x with hx hx,
{ rw [abs_rpow_of_nonneg hx] },
{ rw [abs_of_neg hx, rpow_def_of_neg hx, rpow_def_of_pos (neg_pos.2 hx), log_neg_eq_log,
abs_mul, abs_of_pos (exp_pos _)],
exact mul_le_of_le_one_right (exp_pos _).le (abs_cos_le_one _) }
end | lemma | real.abs_rpow_le_abs_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"abs_mul",
"abs_of_neg",
"abs_of_pos",
"mul_le_of_le_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_rpow_le_exp_log_mul (x y : ℝ) : |x ^ y| ≤ exp (log x * y) | begin
refine (abs_rpow_le_abs_rpow x y).trans _,
by_cases hx : x = 0,
{ by_cases hy : y = 0; simp [hx, hy, zero_le_one] },
{ rw [rpow_def_of_pos (abs_pos.2 hx), log_abs] }
end | lemma | real.abs_rpow_le_exp_log_mul | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"exp",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_rpow_of_nonneg {x y : ℝ} (hx_nonneg : 0 ≤ x) : ‖x ^ y‖ = ‖x‖ ^ y | by { simp_rw real.norm_eq_abs, exact abs_rpow_of_nonneg hx_nonneg, } | lemma | real.norm_rpow_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.norm_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add (hx : 0 < x) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z | by simp only [rpow_def_of_pos hx, mul_add, exp_add] | lemma | real.rpow_add | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"exp_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add' (hx : 0 ≤ x) (h : y + z ≠ 0) : x ^ (y + z) = x ^ y * x ^ z | begin
rcases hx.eq_or_lt with rfl|pos,
{ rw [zero_rpow h, zero_eq_mul],
have : y ≠ 0 ∨ z ≠ 0, from not_and_distrib.1 (λ ⟨hy, hz⟩, h $ hy.symm ▸ hz.symm ▸ zero_add 0),
exact this.imp zero_rpow zero_rpow },
{ exact rpow_add pos _ _ }
end | lemma | real.rpow_add' | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"zero_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add_of_nonneg (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) :
x ^ (y + z) = x ^ y * x ^ z | begin
rcases hy.eq_or_lt with rfl|hy,
{ rw [zero_add, rpow_zero, one_mul] },
exact rpow_add' hx (ne_of_gt $ add_pos_of_pos_of_nonneg hy hz)
end | lemma | real.rpow_add_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_rpow_add {x : ℝ} (hx : 0 ≤ x) (y z : ℝ) : x ^ y * x ^ z ≤ x ^ (y + z) | begin
rcases le_iff_eq_or_lt.1 hx with H|pos,
{ by_cases h : y + z = 0,
{ simp only [H.symm, h, rpow_zero],
calc (0 : ℝ) ^ y * 0 ^ z ≤ 1 * 1 :
mul_le_mul (zero_rpow_le_one y) (zero_rpow_le_one z) (zero_rpow_nonneg z) zero_le_one
... = 1 : by simp },
{ simp [rpow_add', ← H, h] } },
{ si... | lemma | real.le_rpow_add | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_le_mul",
"zero_le_one"
] | For `0 ≤ x`, the only problematic case in the equality `x ^ y * x ^ z = x ^ (y + z)` is for
`x = 0` and `y + z = 0`, where the right hand side is `1` while the left hand side can vanish.
The inequality is always true, though, and given in this lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_sum_of_pos {ι : Type*} {a : ℝ} (ha : 0 < a) (f : ι → ℝ) (s : finset ι) :
a ^ (∑ x in s, f x) = ∏ x in s, a ^ f x | @add_monoid_hom.map_sum ℝ ι (additive ℝ) _ _ ⟨λ x : ℝ, (a ^ x : ℝ), rpow_zero a, rpow_add ha⟩ f s | lemma | real.rpow_sum_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"additive",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_sum_of_nonneg {ι : Type*} {a : ℝ} (ha : 0 ≤ a) {s : finset ι} {f : ι → ℝ}
(h : ∀ x ∈ s, 0 ≤ f x) :
a ^ (∑ x in s, f x) = ∏ x in s, a ^ f x | begin
induction s using finset.cons_induction with i s hi ihs,
{ rw [sum_empty, finset.prod_empty, rpow_zero] },
{ rw forall_mem_cons at h,
rw [sum_cons, prod_cons, ← ihs h.2, rpow_add_of_nonneg ha h.1 (sum_nonneg h.2)] }
end | lemma | real.rpow_sum_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"finset",
"finset.cons_induction",
"finset.prod_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_neg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : x ^ -y = (x ^ y)⁻¹ | by simp only [rpow_def_of_nonneg hx]; split_ifs; simp [*, exp_neg] at * | lemma | real.rpow_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"exp_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_sub {x : ℝ} (hx : 0 < x) (y z : ℝ) : x ^ (y - z) = x ^ y / x ^ z | by simp only [sub_eq_add_neg, rpow_add hx, rpow_neg (le_of_lt hx), div_eq_mul_inv] | lemma | real.rpow_sub | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_sub' {x : ℝ} (hx : 0 ≤ x) {y z : ℝ} (h : y - z ≠ 0) :
x ^ (y - z) = x ^ y / x ^ z | by { simp only [sub_eq_add_neg] at h ⊢, simp only [rpow_add' hx h, rpow_neg hx, div_eq_mul_inv] } | lemma | real.rpow_sub' | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_cpow {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : ((x ^ y : ℝ) : ℂ) = (x : ℂ) ^ (y : ℂ) | by simp only [real.rpow_def_of_nonneg hx, complex.cpow_def, of_real_eq_zero]; split_ifs;
simp [complex.of_real_log hx] | lemma | complex.of_real_cpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"complex.cpow_def",
"complex.of_real_log",
"real.rpow_def_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_cpow_of_nonpos {x : ℝ} (hx : x ≤ 0) (y : ℂ) :
(x : ℂ) ^ y = ((-x) : ℂ) ^ y * exp (π * I * y) | begin
rcases hx.eq_or_lt with rfl|hlt,
{ rcases eq_or_ne y 0 with rfl|hy; simp * },
have hne : (x : ℂ) ≠ 0, from of_real_ne_zero.mpr hlt.ne,
rw [cpow_def_of_ne_zero hne, cpow_def_of_ne_zero (neg_ne_zero.2 hne), ← exp_add, ← add_mul,
log, log, abs.map_neg, arg_of_real_of_neg hlt, ← of_real_neg,
arg_o... | lemma | complex.of_real_cpow_of_nonpos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"eq_or_ne",
"exp",
"exp_add",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cpow_of_ne_zero {z : ℂ} (hz : z ≠ 0) (w : ℂ) :
abs (z ^ w) = abs z ^ w.re / real.exp (arg z * im w) | by rw [cpow_def_of_ne_zero hz, abs_exp, mul_re, log_re, log_im, real.exp_sub,
real.rpow_def_of_pos (abs.pos hz)] | lemma | complex.abs_cpow_of_ne_zero | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.exp",
"real.exp_sub",
"real.rpow_def_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cpow_of_imp {z w : ℂ} (h : z = 0 → w.re = 0 → w = 0) :
abs (z ^ w) = abs z ^ w.re / real.exp (arg z * im w) | begin
rcases ne_or_eq z 0 with hz|rfl; [exact (abs_cpow_of_ne_zero hz w), rw map_zero],
cases eq_or_ne w.re 0 with hw hw,
{ simp [hw, h rfl hw] },
{ rw [real.zero_rpow hw, zero_div, zero_cpow, map_zero],
exact ne_of_apply_ne re hw }
end | lemma | complex.abs_cpow_of_imp | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"eq_or_ne",
"ne_of_apply_ne",
"ne_or_eq",
"real.exp",
"real.zero_rpow",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cpow_le (z w : ℂ) : abs (z ^ w) ≤ abs z ^ w.re / real.exp (arg z * im w) | begin
rcases ne_or_eq z 0 with hz|rfl; [exact (abs_cpow_of_ne_zero hz w).le, rw map_zero],
rcases eq_or_ne w 0 with rfl|hw, { simp },
rw [zero_cpow hw, map_zero],
exact div_nonneg (real.rpow_nonneg_of_nonneg le_rfl _) (real.exp_pos _).le
end | lemma | complex.abs_cpow_le | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"div_nonneg",
"eq_or_ne",
"le_rfl",
"ne_or_eq",
"real.exp",
"real.exp_pos",
"real.rpow_nonneg_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cpow_real (x : ℂ) (y : ℝ) : abs (x ^ (y : ℂ)) = x.abs ^ y | by rcases eq_or_ne x 0 with rfl|hx; [rcases eq_or_ne y 0 with rfl|hy, skip];
simp [*, abs_cpow_of_ne_zero] | lemma | complex.abs_cpow_real | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cpow_inv_nat (x : ℂ) (n : ℕ) : abs (x ^ (n⁻¹ : ℂ)) = x.abs ^ (n⁻¹ : ℝ) | by rw ← abs_cpow_real; simp [-abs_cpow_real] | lemma | complex.abs_cpow_inv_nat | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cpow_eq_rpow_re_of_pos {x : ℝ} (hx : 0 < x) (y : ℂ) : abs (x ^ y) = x ^ y.re | by rw [abs_cpow_of_ne_zero (of_real_ne_zero.mpr hx.ne'), arg_of_real_of_nonneg hx.le, zero_mul,
real.exp_zero, div_one, abs_of_nonneg hx.le] | lemma | complex.abs_cpow_eq_rpow_re_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"abs_of_nonneg",
"div_one",
"real.exp_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cpow_eq_rpow_re_of_nonneg {x : ℝ} (hx : 0 ≤ x) {y : ℂ} (hy : re y ≠ 0) :
abs (x ^ y) = x ^ re y | begin
rcases hx.eq_or_lt with rfl|hlt,
{ rw [of_real_zero, zero_cpow, map_zero, real.zero_rpow hy],
exact ne_of_apply_ne re hy },
{ exact abs_cpow_eq_rpow_re_of_pos hlt y }
end | lemma | complex.abs_cpow_eq_rpow_re_of_nonneg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"ne_of_apply_ne",
"real.zero_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_mul {x : ℝ} (hx : 0 ≤ x) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z | by rw [← complex.of_real_inj, complex.of_real_cpow (rpow_nonneg_of_nonneg hx _),
complex.of_real_cpow hx, complex.of_real_mul, complex.cpow_mul, complex.of_real_cpow hx];
simp only [(complex.of_real_mul _ _).symm, (complex.of_real_log hx).symm,
complex.of_real_im, neg_lt_zero, pi_pos, le_of_lt pi_pos] | lemma | real.rpow_mul | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"complex.cpow_mul",
"complex.of_real_cpow",
"complex.of_real_im",
"complex.of_real_inj",
"complex.of_real_log",
"complex.of_real_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add_int {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℤ) : x ^ (y + n) = x ^ y * x ^ n | by rw [rpow_def, complex.of_real_add, complex.cpow_add _ _ (complex.of_real_ne_zero.mpr hx),
complex.of_real_int_cast, complex.cpow_int_cast, ← complex.of_real_zpow, mul_comm,
complex.of_real_mul_re, ← rpow_def, mul_comm] | lemma | real.rpow_add_int | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"complex.cpow_add",
"complex.cpow_int_cast",
"complex.of_real_add",
"complex.of_real_int_cast",
"complex.of_real_mul_re",
"complex.of_real_zpow",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add_nat {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y + n) = x ^ y * x ^ n | by simpa using rpow_add_int hx y n | lemma | real.rpow_add_nat | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_sub_int {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℤ) : x ^ (y - n) = x ^ y / x ^ n | by simpa using rpow_add_int hx y (-n) | lemma | real.rpow_sub_int | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_sub_nat {x : ℝ} (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y - n) = x ^ y / x ^ n | by simpa using rpow_sub_int hx y n | lemma | real.rpow_sub_nat | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add_one {x : ℝ} (hx : x ≠ 0) (y : ℝ) : x ^ (y + 1) = x ^ y * x | by simpa using rpow_add_nat hx y 1 | lemma | real.rpow_add_one | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_sub_one {x : ℝ} (hx : x ≠ 0) (y : ℝ) : x ^ (y - 1) = x ^ y / x | by simpa using rpow_sub_nat hx y 1 | lemma | real.rpow_sub_one | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_int_cast (x : ℝ) (n : ℤ) : x ^ (n : ℝ) = x ^ n | by simp only [rpow_def, ← complex.of_real_zpow, complex.cpow_int_cast,
complex.of_real_int_cast, complex.of_real_re] | lemma | real.rpow_int_cast | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"complex.cpow_int_cast",
"complex.of_real_int_cast",
"complex.of_real_re",
"complex.of_real_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_nat_cast (x : ℝ) (n : ℕ) : x ^ (n : ℝ) = x ^ n | by simpa using rpow_int_cast x n | lemma | real.rpow_nat_cast | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_two (x : ℝ) : x ^ (2 : ℝ) = x ^ 2 | by { rw ← rpow_nat_cast, simp only [nat.cast_bit0, nat.cast_one] } | lemma | real.rpow_two | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"nat.cast_bit0",
"nat.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_neg_one (x : ℝ) : x ^ (-1 : ℝ) = x⁻¹ | begin
suffices H : x ^ ((-1 : ℤ) : ℝ) = x⁻¹, by rwa [int.cast_neg, int.cast_one] at H,
simp only [rpow_int_cast, zpow_one, zpow_neg],
end | lemma | real.rpow_neg_one | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"int.cast_neg",
"int.cast_one",
"zpow_neg",
"zpow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_rpow {x y z : ℝ} (h : 0 ≤ x) (h₁ : 0 ≤ y) : (x*y)^z = x^z * y^z | begin
iterate 3 { rw real.rpow_def_of_nonneg }, split_ifs; simp * at *,
{ have hx : 0 < x,
{ cases lt_or_eq_of_le h with h₂ h₂, { exact h₂ },
exfalso, apply h_2, exact eq.symm h₂ },
have hy : 0 < y,
{ cases lt_or_eq_of_le h₁ with h₂ h₂, { exact h₂ },
exfalso, apply h_3, exact eq.symm h₂ },
... | lemma | real.mul_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"exp_add",
"real.rpow_def_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_rpow (hx : 0 ≤ x) (y : ℝ) : (x⁻¹)^y = (x^y)⁻¹ | by simp only [← rpow_neg_one, ← rpow_mul hx, mul_comm] | lemma | real.inv_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_rpow (hx : 0 ≤ x) (hy : 0 ≤ y) (z : ℝ) : (x / y) ^ z = x^z / y^z | by simp only [div_eq_mul_inv, mul_rpow hx (inv_nonneg.2 hy), inv_rpow hy] | lemma | real.div_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_rpow {x : ℝ} (hx : 0 < x) (y : ℝ) : log (x^y) = y * (log x) | begin
apply exp_injective,
rw [exp_log (rpow_pos_of_pos hx y), ← exp_log hx, mul_comm, rpow_def_of_pos (exp_pos (log x)) y],
end | lemma | real.log_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow (hx : 0 ≤ x) (hxy : x < y) (hz : 0 < z) : x^z < y^z | begin
rw le_iff_eq_or_lt at hx, cases hx,
{ rw [← hx, zero_rpow (ne_of_gt hz)], exact rpow_pos_of_pos (by rwa ← hx at hxy) _ },
rw [rpow_def_of_pos hx, rpow_def_of_pos (lt_trans hx hxy), exp_lt_exp],
exact mul_lt_mul_of_pos_right (log_lt_log hx hxy) hz
end | lemma | real.rpow_lt_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"le_iff_eq_or_lt",
"mul_lt_mul_of_pos_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow {x y z: ℝ} (h : 0 ≤ x) (h₁ : x ≤ y) (h₂ : 0 ≤ z) : x^z ≤ y^z | begin
rcases eq_or_lt_of_le h₁ with rfl|h₁', { refl },
rcases eq_or_lt_of_le h₂ with rfl|h₂', { simp },
exact le_of_lt (rpow_lt_rpow h h₁' h₂')
end | lemma | real.rpow_le_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"eq_or_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow_iff (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ^ z < y ^ z ↔ x < y | ⟨lt_imp_lt_of_le_imp_le $ λ h, rpow_le_rpow hy h (le_of_lt hz), λ h, rpow_lt_rpow hx h hz⟩ | lemma | real.rpow_lt_rpow_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_iff (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 < z) : x ^ z ≤ y ^ z ↔ x ≤ y | le_iff_le_iff_lt_iff_lt.2 $ rpow_lt_rpow_iff hy hx hz | lemma | real.rpow_le_rpow_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_rpow_inv_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) :
x ≤ y ^ z⁻¹ ↔ y ≤ x ^ z | begin
have hz' : 0 < -z := by rwa [lt_neg, neg_zero],
have hxz : 0 < x ^ (-z) := real.rpow_pos_of_pos hx _,
have hyz : 0 < y ^ z⁻¹ := real.rpow_pos_of_pos hy _,
rw [←real.rpow_le_rpow_iff hx.le hyz.le hz', ←real.rpow_mul hy.le],
simp only [ne_of_lt hz, real.rpow_neg_one, mul_neg, inv_mul_cancel, ne.def, not_f... | lemma | real.le_rpow_inv_iff_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"inv_mul_cancel",
"le_inv",
"mul_neg",
"real.rpow_neg_one",
"real.rpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_rpow_inv_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) :
x < y ^ z⁻¹ ↔ y < x ^ z | begin
have hz' : 0 < -z := by rwa [lt_neg, neg_zero],
have hxz : 0 < x ^ (-z) := real.rpow_pos_of_pos hx _,
have hyz : 0 < y ^ z⁻¹ := real.rpow_pos_of_pos hy _,
rw [←real.rpow_lt_rpow_iff hx.le hyz.le hz', ←real.rpow_mul hy.le],
simp only [ne_of_lt hz, real.rpow_neg_one, mul_neg, inv_mul_cancel, ne.def, not_f... | lemma | real.lt_rpow_inv_iff_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"inv_mul_cancel",
"lt_inv",
"mul_neg",
"real.rpow_neg_one",
"real.rpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_inv_lt_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) :
x ^ z⁻¹ < y ↔ y ^ z < x | begin
convert lt_rpow_inv_iff_of_neg (real.rpow_pos_of_pos hx _) (real.rpow_pos_of_pos hy _) hz;
simp [←real.rpow_mul hx.le, ←real.rpow_mul hy.le, ne_of_lt hz],
end | lemma | real.rpow_inv_lt_iff_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.rpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_inv_le_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) :
x ^ z⁻¹ ≤ y ↔ y ^ z ≤ x | begin
convert le_rpow_inv_iff_of_neg (real.rpow_pos_of_pos hx _) (real.rpow_pos_of_pos hy _) hz;
simp [←real.rpow_mul hx.le, ←real.rpow_mul hy.le, ne_of_lt hz],
end | lemma | real.rpow_inv_le_iff_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.rpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow_of_exponent_lt (hx : 1 < x) (hyz : y < z) : x^y < x^z | begin
repeat {rw [rpow_def_of_pos (lt_trans zero_lt_one hx)]},
rw exp_lt_exp, exact mul_lt_mul_of_pos_left hyz (log_pos hx),
end | lemma | real.rpow_lt_rpow_of_exponent_lt | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_lt_mul_of_pos_left",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_of_exponent_le (hx : 1 ≤ x) (hyz : y ≤ z) : x^y ≤ x^z | begin
repeat {rw [rpow_def_of_pos (lt_of_lt_of_le zero_lt_one hx)]},
rw exp_le_exp, exact mul_le_mul_of_nonneg_left hyz (log_nonneg hx),
end | lemma | real.rpow_le_rpow_of_exponent_le | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_le_mul_of_nonneg_left",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_left_iff (hx : 1 < x) : x ^ y ≤ x ^ z ↔ y ≤ z | begin
have x_pos : 0 < x := lt_trans zero_lt_one hx,
rw [←log_le_log (rpow_pos_of_pos x_pos y) (rpow_pos_of_pos x_pos z),
log_rpow x_pos, log_rpow x_pos, mul_le_mul_right (log_pos hx)],
end | lemma | real.rpow_le_rpow_left_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_le_mul_right",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow_left_iff (hx : 1 < x) : x ^ y < x ^ z ↔ y < z | by rw [lt_iff_not_le, rpow_le_rpow_left_iff hx, lt_iff_not_le] | lemma | real.rpow_lt_rpow_left_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"lt_iff_not_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow_of_exponent_gt (hx0 : 0 < x) (hx1 : x < 1) (hyz : z < y) :
x^y < x^z | begin
repeat {rw [rpow_def_of_pos hx0]},
rw exp_lt_exp, exact mul_lt_mul_of_neg_left hyz (log_neg hx0 hx1),
end | lemma | real.rpow_lt_rpow_of_exponent_gt | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_lt_mul_of_neg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_of_exponent_ge (hx0 : 0 < x) (hx1 : x ≤ 1) (hyz : z ≤ y) :
x^y ≤ x^z | begin
repeat {rw [rpow_def_of_pos hx0]},
rw exp_le_exp, exact mul_le_mul_of_nonpos_left hyz (log_nonpos (le_of_lt hx0) hx1),
end | lemma | real.rpow_le_rpow_of_exponent_ge | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_le_mul_of_nonpos_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_left_iff_of_base_lt_one (hx0 : 0 < x) (hx1 : x < 1) :
x ^ y ≤ x ^ z ↔ z ≤ y | begin
rw [←log_le_log (rpow_pos_of_pos hx0 y) (rpow_pos_of_pos hx0 z),
log_rpow hx0, log_rpow hx0, mul_le_mul_right_of_neg (log_neg hx0 hx1)],
end | lemma | real.rpow_le_rpow_left_iff_of_base_lt_one | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_le_mul_right_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_rpow_left_iff_of_base_lt_one (hx0 : 0 < x) (hx1 : x < 1) :
x ^ y < x ^ z ↔ z < y | by rw [lt_iff_not_le, rpow_le_rpow_left_iff_of_base_lt_one hx0 hx1, lt_iff_not_le] | lemma | real.rpow_lt_rpow_left_iff_of_base_lt_one | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"lt_iff_not_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_one {x z : ℝ} (hx1 : 0 ≤ x) (hx2 : x < 1) (hz : 0 < z) : x^z < 1 | by { rw ← one_rpow z, exact rpow_lt_rpow hx1 hx2 hz } | lemma | real.rpow_lt_one | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_one {x z : ℝ} (hx1 : 0 ≤ x) (hx2 : x ≤ 1) (hz : 0 ≤ z) : x^z ≤ 1 | by { rw ← one_rpow z, exact rpow_le_rpow hx1 hx2 hz } | lemma | real.rpow_le_one | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_one_of_one_lt_of_neg {x z : ℝ} (hx : 1 < x) (hz : z < 0) : x^z < 1 | by { convert rpow_lt_rpow_of_exponent_lt hx hz, exact (rpow_zero x).symm } | lemma | real.rpow_lt_one_of_one_lt_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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